Abstract Euler characteristic is typically thought of as an alternating sum of ranks of homology groups, but it is often better to regard it as a generalization of cardinality or measure, as suggested by the formula χ(X ∪ Y) = χ(X) + χ(Y) - χ(X ∩ Y). This point of view leads to useful notions of Euler characteristic for structures such as groupoids, posets, categories, datatypes (in computer science), and classes of space not amenable to the usual methods of algebraic topology, such as certain spaces important in complex dynamics. Some of these theories of Euler characteristic are well-established, while others are currently being developed. I will give an overview.
Slides In this pdf file (122KB).
Errata On p.6, the calculation of the Euler characteristic of [0, 1]2 is wrong. It should follow the same pattern as the calculation of the Euler characteristic of [0, 1]. Also, the picture on p.9 can't possibly be the underlying graph of a category, as there's no way to define a composition on it. The same goes for the groupoid on p.15. Sorry.
Following some questions, I should also clarify that the discussion of polyconvex sets ends at the end of page 5. On page 6, for instance, we're talking about general topological spaces.